Partial differential equation

Definition

A partial differential equation is an equation that involves an unknown function of ${\displaystyle n\ge 2}$ variables and (some of) its partial derivatives. That is, for the unknown function $${\displaystyle u:U\to \mathbb{R},}$$ of variables ${\displaystyle x=(x_1,\dots ,x_n)}$ belonging to the open subset ${\displaystyle U}$ of ${\displaystyle {\mathbb{R}}^n}$, the ${\displaystyle k^{th}}$-order partial differential equation is defined as $${\displaystyle F[D^{k}u,D^{k-1}u,\dots ,Du,u,x]=0,}$$ where $${\displaystyle F:{\mathbb{R}}^{n^k}\times {\mathbb{R}}^{n^{k-1}}\cdots \times {\mathbb{R}}^n\times \mathbb{R}\times U\to \mathbb{R},}$$ and ${\displaystyle D}$ is the derivative operator.

Notation

Working in ${\displaystyle {\mathbb{R}}^n}$, the partial derivatives of a function ${\displaystyle u}$ can be denoted by $${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}}$$ or with a subscript $${\displaystyle u_{x_i}.}$$

For multiple derivatives, multi-index notation can be used. Thus if ${\displaystyle \alpha =({\alpha }_1,\dots ,{\alpha }_n)}$, the length of ${\displaystyle \alpha }$ is denoted by ${\displaystyle |\alpha |={\alpha }_1+\cdots +{\alpha }_n}$ and the iterated partial is denoted by $${\displaystyle D^{\alpha }u=\frac{{\mathrm{\partial }}^{k}u}{\mathrm{\partial }{x}_1^{{\alpha }_1}\cdots \mathrm{\partial }{x}_n^{{\alpha }_n}}.}$$

In the above definition of a partial differential equation the powers of ${\displaystyle D}$ are the tensors whose components are the partial derivatives of ${\displaystyle u}$; for example ${\displaystyle D^{k}u}$ is a tensor having ${\displaystyle n^k}$ components that that are an arrangement of the set ${\displaystyle \{D^{\alpha }u\mid |\alpha |=k\}}$ after accounting for commutativity of partial derivatives.

The Greek letter Δ denotes the Laplace operator; if u is a function of n variables, then $${\displaystyle \mathrm{\Delta }u=u_{11}+u_{22}+\cdots +u_{nn}.}$$ In the physics literature, the Laplace operator is often denoted by ∇²; in the mathematics literature, ∇²u may also denote the Hessian matrix of u, which is here denoted by ${\displaystyle D^{2}u}$.

Classification

Linear and nonlinear equations

A PDE is called linear if it is linear in the unknown and its derivatives. For example, for a function u of x and y, a second order linear PDE is of the form $${\displaystyle a_1(x,y)u_{xx}+a_2(x,y)u_{xy}+a_3(x,y)u_{yx}+a_4(x,y)u_{yy}+a_5(x,y)u_x+a_6(x,y)u_y+a_7(x,y)u=f(x,y)}$$ where a_i and f are functions of the independent variables x and y only. (Often the mixed-partial derivatives u_xy and u_yx will be equated, but this is not required for the discussion of linearity.) If the a_i are constants (independent of x and y) then the PDE is called linear with constant coefficients. If f is zero everywhere then the linear PDE is homogeneous, otherwise it is inhomogeneous. (This is separate from asymptotic homogenization, which studies the effects of high-frequency oscillations in the coefficients upon solutions to PDEs.)

Nearest to linear PDEs are semi-linear PDEs, where only the highest order derivatives appear as linear terms, with coefficients that are functions of the independent variables. The lower order derivatives and the unknown function may appear arbitrarily. For example, a general second order semi-linear PDE in two variables is $${\displaystyle a_1(x,y)u_{xx}+a_2(x,y)u_{xy}+a_3(x,y)u_{yx}+a_4(x,y)u_{yy}+f(u_x,u_y,u,x,y)=0}$$

In a quasilinear PDE the highest order derivatives likewise appear only as linear terms, but with coefficients possibly functions of the unknown and lower-order derivatives: $${\displaystyle a_1(u_x,u_y,u,x,y)u_{xx}+a_2(u_x,u_y,u,x,y)u_{xy}+a_3(u_x,u_y,u,x,y)u_{yx}+a_4(u_x,u_y,u,x,y)u_{yy}+f(u_x,u_y,u,x,y)=0}$$ Many of the fundamental PDEs in physics are quasilinear, such as the Einstein equations of general relativity and the Navier–Stokes equations describing fluid motion.

A PDE without any linearity properties is called fully nonlinear, and possesses nonlinearities on one or more of the highest-order derivatives. An example is the Monge–Ampère equation, which arises in differential geometry.

Second order equations

The elliptic/parabolic/hyperbolic classification provides a guide to appropriate initial- and boundary conditions and to the smoothness of the solutions. Assuming u_xy = u_yx, the general linear second-order PDE in two independent variables has the form $${\displaystyle A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}+\cdots \text{(lower order terms)}=0,}$$ where the coefficients A, B, C... may depend upon x and y. If A² + B² + C² > 0 over a region of the xy-plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section: $${\displaystyle A{x}^2+2Bxy+C{y}^2+\cdots =0.}$$

More precisely, replacing ∂_x by X, and likewise for other variables (formally this is done by a Fourier transform), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a homogeneous polynomial, here a quadratic form) being most significant for the classification.

Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant B² − 4AC, the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by B² − AC due to the convention of the xy term being 2B rather than B; formally, the discriminant (of the associated quadratic form) is (2B)² − 4AC = 4(B² − AC), with the factor of 4 dropped for simplicity.

1. B² − AC < 0 (elliptic partial differential equation): Solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of Laplace's equation are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where x < 0. By a change of variables, the equation can always be expressed in the form: $${\displaystyle u_{xx}+u_{yy}+\cdots =0,}$$where x and y correspond to changed variables. This justifies Laplace equation as an example of this type.
2. B² − AC = 0 (parabolic partial differential equation): Equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where x = 0. By change of variables, the equation can always be expressed in the form: $${\displaystyle u_{xx}+\cdots =0,}$$where x correspond to changed variables. This justifies the heat equation, which is of the form $u_t-u_{xx}+\cdots =0$, as an example of this type.
3. B² − AC > 0 (hyperbolic partial differential equation): hyperbolic equations retain any discontinuities of functions or derivatives in the initial data. An example is the wave equation. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where x > 0. By change of variables, the equation can always be expressed in the form: $${\displaystyle u_{xx}-u_{yy}+\cdots =0,}$$where x and y correspond to changed variables. This justifies the wave equation as an example of this type.

If there are n independent variables x₁, x₂ , …, x_n, a general linear partial differential equation of second order has the form $${\displaystyle Lu=\sum _{i=1}^n\sum _{j=1}^n{a}_{i,j}\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}+\text{lower-order terms}=0.}$$

The classification depends upon the signature of the eigenvalues of the coefficient matrix a_i,j.

1. Elliptic: the eigenvalues are all positive or all negative.
2. Parabolic: the eigenvalues are all positive or all negative, except one that is zero.
3. Hyperbolic: there is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.
4. Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues.

The theory of elliptic, parabolic, and hyperbolic equations have been studied for centuries, largely centered around or based upon the standard examples of the Laplace equation, the heat equation, and the wave equation.

However, the classification only depends on linearity of the second-order terms and is therefore applicable to semi- and quasilinear PDEs as well. The basic types also extend to hybrids such as the Euler–Tricomi equation; varying from elliptic to hyperbolic for different regions of the domain, as well as higher-order PDEs, but such knowledge is more specialized.

Systems of first-order equations and characteristic surfaces

The classification of partial differential equations can be extended to systems of first-order equations, where the unknown u is now a vector with m components, and the coefficient matrices A_ν are m by m matrices for ν = 1, 2, …, n. The partial differential equation takes the form $${\displaystyle Lu=\sum _{\nu =1}^n{A}_{\nu }\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_{\nu }}+B=0,}$$ where the coefficient matrices A_ν and the vector B may depend upon x and u. If a hypersurface S is given in the implicit form $${\displaystyle \phi (x_1,x_2,\dots ,x_n)=0,}$$ where φ has a non-zero gradient, then S is a characteristic surface for the operator L at a given point if the characteristic form vanishes: $${\displaystyle Q\left(\frac{\mathrm{\partial }\phi }{\mathrm{\partial }{x}_1},\dots ,\frac{\mathrm{\partial }\phi }{\mathrm{\partial }{x}_n}\right)=det\left[\sum _{\nu =1}^n{A}_{\nu }\frac{\mathrm{\partial }\phi }{\mathrm{\partial }{x}_{\nu }}\right]=0.}$$

The geometric interpretation of this condition is as follows: if data for u are prescribed on the surface S, then it may be possible to determine the normal derivative of u on S from the differential equation. If the data on S and the differential equation determine the normal derivative of u on S, then S is non-characteristic. If the data on S and the differential equation do not determine the normal derivative of u on S, then the surface is characteristic, and the differential equation restricts the data on S: the differential equation is internal to S.

1. A first-order system Lu = 0 is elliptic if no surface is characteristic for L: the values of u on S and the differential equation always determine the normal derivative of u on S.
2. A first-order system is hyperbolic at a point if there is a spacelike surface S with normal ξ at that point. This means that, given any non-trivial vector η orthogonal to ξ, and a scalar multiplier λ, the equation Q(λξ + η) = 0 has m real roots λ₁, λ₂, …, λ_m. The system is strictly hyperbolic if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form Q(ζ) = 0 defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has nm sheets, and the axis ζ = λξ runs inside these sheets: it does not intersect any of them. But when displaced from the origin by η, this axis intersects every sheet. In the elliptic case, the normal cone has no real sheets.

Analytical solutions

Separation of variables

Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. This technique rests on a feature of solutions to differential equations: if one can find any solution that solves the equation and satisfies the boundary conditions, then it is the solution (this also applies to ODEs). We assume as an ansatz that the dependence of a solution on the parameters space and time can be written as a product of terms that each depend on a single parameter, and then see if this can be made to solve the problem.

In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ordinary differential equation if in one variable – these are in turn easier to solve.

This is possible for simple PDEs, which are called separable partial differential equations, and the domain is generally a rectangle (a product of intervals). Separable PDEs correspond to diagonal matrices – thinking of "the value for fixed x" as a coordinate, each coordinate can be understood separately.

This generalizes to the method of characteristics, and is also used in integral transforms.

Method of characteristics

The characteristic surface in n = 2-dimensional space is called a characteristic curve. In special cases, one can find characteristic curves on which the first-order PDE reduces to an ODE – changing coordinates in the domain to straighten these curves allows separation of variables, and is called the method of characteristics.

More generally, applying the method to first-order PDEs in higher dimensions, one may find characteristic surfaces.

Integral transform

An integral transform may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator.

An important example of this is Fourier analysis, which diagonalizes the heat equation using the eigenbasis of sinusoidal waves.

If the domain is finite or periodic, an infinite sum of solutions such as a Fourier series is appropriate, but an integral of solutions such as a Fourier integral is generally required for infinite domains. The solution for a point source for the heat equation given above is an example of the use of a Fourier integral.

Change of variables

Often a PDE can be reduced to a simpler form with a known solution by a suitable change of variables. For example, the Black–Scholes equation $${\displaystyle \frac{\mathrm{\partial }V}{\mathrm{\partial }t}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }V}{\mathrm{\partial }S}-rV=0}$$ is reducible to the heat equation $${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }\tau }=\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}$$ by the change of variables $${\displaystyle \begin{array}{cc}V(S,t)& =v(x,\tau ),\\ x& =\ln \left(S\right),\\ \tau & =\frac{1}{2}{\sigma }^2(T-t),\\ v(x,\tau )& =e^{-\alpha x-\beta \tau }u(x,\tau ).\end{array}}$$

Fundamental solution

Inhomogeneous equations can often be solved (for constant coefficient PDEs, always be solved) by finding the fundamental solution (the solution for a point source ${\displaystyle P(D)u=\delta }$), then taking the convolution with the boundary conditions to get the solution.

This is analogous in signal processing to understanding a filter by its impulse response.

Superposition principle

The superposition principle applies to any linear system, including linear systems of PDEs. A common visualization of this concept is the interaction of two waves in phase being combined to result in a greater amplitude, for example sin x + sin x = 2 sin x. The same principle can be observed in PDEs where the solutions may be real or complex and additive. If u₁ and u₂ are solutions of linear PDE in some function space R, then u = c₁u₁ + c₂u₂ with any constants c₁ and c₂ are also a solution of that PDE in the same function space.

Methods for non-linear equations

There are no generally applicable analytical methods to solve nonlinear PDEs. Still, existence and uniqueness results (such as the Cauchy–Kowalevski theorem) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of analysis).

Nevertheless, some techniques can be used for several types of equations. The h-principle is the most powerful method to solve underdetermined equations. The Riquier–Janet theory is an effective method for obtaining information about many analytic overdetermined systems.

The method of characteristics can be used in some very special cases to solve nonlinear partial differential equations.

In some cases, a PDE can be solved via perturbation analysis in which the solution is considered to be a correction to an equation with a known solution. Alternatives are numerical analysis techniques from simple finite difference schemes to the more mature multigrid and finite element methods. Many interesting problems in science and engineering are solved in this way using computers, sometimes high performance supercomputers.

Lie group method

From 1870 Sophus Lie's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called Lie groups, be referred, to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration. He also emphasized the subject of transformations of contact.

A general approach to solving PDEs uses the symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions (Lie theory). Continuous group theory, Lie algebras and differential geometry are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform and finally finding exact analytic solutions to the PDE.

Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.

Semi-analytical methods

The Adomian decomposition method, the Lyapunov artificial small parameter method, and his homotopy perturbation method are all special cases of the more general homotopy analysis method. These are series expansion methods, and except for the Lyapunov method, are independent of small physical parameters as compared to the well known perturbation theory, thus giving these methods greater flexibility and solution generality.

Numerical solutions

The three most widely used numerical methods to solve PDEs are the finite element method (FEM), finite volume methods (FVM) and finite difference methods (FDM), as well other kind of methods called meshfree methods, which were made to solve problems where the aforementioned methods are limited. The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order version hp-FEM. Other hybrid versions of FEM and Meshfree methods include the generalized finite element method (GFEM), extended finite element method (XFEM), spectral finite element method (SFEM), meshfree finite element method, discontinuous Galerkin finite element method (DGFEM), element-free Galerkin method (EFGM), interpolating element-free Galerkin method (IEFGM), etc.

Finite element method

The finite element method (FEM) (its practical application often known as finite element analysis (FEA)) is a numerical technique for approximating solutions of partial differential equations (PDE) as well as of integral equations using a finite set of functions. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge–Kutta, etc.

Finite difference method

Finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.

Finite volume method

Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, surface integrals in a partial differential equation that contain a divergence term are converted to volume integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods conserve mass by design.

Neural networks

Physics-informed neural networks have been used to solve partial differential equations in both forward and inverse problems in a data driven manner. One example is the reconstructing fluid flow governed by the Navier-Stokes equations. Using physics informed neural networks does not require the often expensive mesh generation that conventional CFD methods rely on. It is evident that geometric and physical constraints have a synergistic effect on neural PDE surrogates, thereby enhancing their efficacy in predicting stable and super long rollouts.

Weak solutions

Weak solutions are functions that satisfy the PDE, yet in other meanings than regular sense. The meaning for this term may differ with context, and one of the most commonly used definitions is based on the notion of distributions.

An example for the definition of a weak solution is as follows:

Consider the boundary-value problem given by: $${\displaystyle \begin{array}{cc}Lu& =f\text{in }U,\\ u& =0\text{on }\mathrm{\partial }U,\end{array}}$$ where $${\displaystyle Lu=-\sum _{i,j}{\mathrm{\partial }}_j(a^{ij}{\mathrm{\partial }}_{i}u)+\sum _i{b}^i{\mathrm{\partial }}_{i}u+cu}$$ denotes a second-order partial differential operator in divergence form.

We say a ${\displaystyle u\in H_0^1(U)}$ is a weak solution if $${\displaystyle {\int }_U[\sum _{i,j}{a}^{ij}({\mathrm{\partial }}_{i}u)({\mathrm{\partial }}_{j}v)+\sum _i{b}^i({\mathrm{\partial }}_{i}u)v+cuv]dx={\int }_{U}fvdx}$$ for every ${\displaystyle v\in H_0^1(U)}$, which can be derived by a formal integral by parts.

An example for a weak solution is as follows: $${\displaystyle \varphi (x)=\frac{1}{4\pi }\frac{1}{|x|}}$$ is a weak solution satisfying $${\displaystyle {\mathrm{\nabla }}^2\varphi =\delta \text{ in }{R}^3}$$ in distributional sense, as formally, $${\displaystyle {\int }_{R^3}{\mathrm{\nabla }}^2\varphi (x)\psi (x)dx={\int }_{R^3}\varphi (x){\mathrm{\nabla }}^2\psi (x)dx=\psi (0)\text{ for }\psi \in C_c^{\mathrm{\infty }}(R^3).}$$

Theoretical studies

In pure mathematics, the theoretical studies of PDEs focus on the criteria for a solution to exist and the properties of a solution while finding its formula is often secondary.

Well-posedness

Well-posedness refers to a common schematic package of information about a PDE. To say that a PDE is well-posed, one must have:

• an existence and uniqueness theorem, asserting that by the prescription of some freely chosen functions, one can single out one specific solution of the PDE
• by continuously changing the free choices, one continuously changes the corresponding solution

This is, by the necessity of being applicable to several different PDE, somewhat vague. The requirement of "continuity", in particular, is ambiguous, since there are usually many inequivalent means by which it can be rigorously defined. It is, however, somewhat unusual to study a PDE without specifying a way in which it is well-posed.

Regularity

Regularity refers to the integrability and differentiability of weak solutions, which can often be represented by Sobolev spaces.

This problem arise due to the difficulty in searching for classical solutions. Researchers often tend to find weak solutions at first and then find out whether it is smooth enough to be qualified as a classical solution.

Results from functional analysis are often used in this field of study.